We study an infinite-horizon consumption, portfolio, and early-retirement problem for a wage earner who receives labor income while working but bears a constant utility cost of labor. Retirement is irreversible and removes both labor income and work disutility. In addition, retirement is feasible only after the agent has accumulated a prescribed level of financial wealth. We solve the problem under constant relative risk aversion using a dual martingale method. The post-retirement problem reduces to the standard Merton problem, while the pre-retirement problem becomes an optimal stopping problem with a target-induced obstacle. We derive an explicit dual value function, characterize the free boundary, recover the primal value, and obtain closed-form consumption, portfolio, and retirement policies. The solution exhibits a sharp slack–binding dichotomy: small targets do not affect the classical disutility retirement policy, whereas large targets become the effective retirement threshold and reshape both consumption and risky investment before retirement.
Kim et al. (Thu,) studied this question.